On Polynomially Integrable Domains in Euclidean Spaces

Let D be a bounded domain in ℝⁿ, with smooth boundary. Denote VD(ω,t),ω∈Sⁿ⁻¹,t∈ℝ, the Radon transform of the characteristic function χ_D of the domain D, i.e., the (n − 1)-dimensional volume of the intersection D with the hyperplane { x∈ ℝⁿ: < ω, x> = t}. If the domain D is an ellipsoid, then the function V_D is algebraic and if, in addition, the dimension n is odd, then V (ω, t) is a polynomial with respect to t. Whether odd-dimensional ellipsoids are the only bounded smooth domains with such a property? The article is devoted to partial verification and discussion of this question.